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The Tolman–Oppenheimer–Volkoff limit (or TOV limit) is an upper bound to the
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
of cold, non-rotating
neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
s, analogous to the Chandrasekhar limit for
white dwarf A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
stars. Stars more massive than the TOV limit collapse into a
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
. The original calculation in 1939, which neglected complications such as nuclear forces between neutrons, placed this limit at approximately 0.7
solar mass The solar mass () is a frequently used unit of mass in astronomy, equal to approximately . It is approximately equal to the mass of the Sun. It is often used to indicate the masses of other stars, as well as stellar clusters, nebulae, galaxie ...
es (). Later, more refined analyses have resulted in larger values. Theoretical work in 1996 placed the limit at approximately 1.5 to 3.0 , corresponding to an original stellar mass of 15 to 20 ; additional work in the same year gave a more precise range of 2.2 to 2.9 . Data from
GW170817 GW170817 was a gravitational wave (GW) observed by the LIGO and Virgo detectors on 17 August 2017, originating within the shell elliptical galaxy NGC 4993, about 144 million light years away. The wave was produced by the last moments of the in ...
, the first gravitational wave observation attributed to merging neutron stars (thought to have collapsed into a black hole within a few seconds after merging) placed the limit in the range of 2.01 to 2.17 . In the case of a rigidly spinning neutron star, meaning that different levels in the interior of the star all rotate at the same rate, the mass limit is thought to increase by up to 18–20%.


History

The idea that there should be an absolute upper limit for the mass of a cold (as distinct from thermal pressure supported) self-gravitating body dates back to the 1932 work of
Lev Landau Lev Davidovich Landau (; 22 January 1908 – 1 April 1968) was a Soviet physicist who made fundamental contributions to many areas of theoretical physics. He was considered as one of the last scientists who were universally well-versed and ma ...
, based on the Pauli exclusion principle. Pauli's principle shows that the
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
ic particles in sufficiently compressed matter would be forced into energy states so high that their
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
contribution would become negligible when compared with the relativistic kinetic contribution (RKC). RKC is determined just by the relevant quantum wavelength , which would be of the order of the mean interparticle separation. In terms of
Planck units In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: ''Speed of light, c'', ''Gravitational constant, G'', ''Reduced Planck constant, ħ ...
, with the
reduced Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
, the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
, and the
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
all set equal to one, there will be a corresponding
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
given roughly by At the upper mass limit, that pressure will equal the pressure needed to resist gravity. The pressure to resist gravity for a body of mass will be given according to the
virial theorem In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by a conservative force (where the work done is independent of path), with ...
roughly by where is the density. This will be given by , where is the relevant mass per particle. It can be seen that the wavelength cancels out so that one obtains an approximate mass limit formula of the very simple form In this relationship, can be taken to be given roughly by the proton mass. This even applies in the
white dwarf A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
case (that of the Chandrasekhar limit) for which the fermionic particles providing the pressure are electrons. This is because the mass density is provided by the nuclei in which the neutrons are at most about as numerous as the protons. Likewise, the protons, for charge neutrality, must be exactly as numerous as the electrons outside. In the case of neutron stars this limit was first worked out by
J. Robert Oppenheimer J. Robert Oppenheimer (born Julius Robert Oppenheimer ; April 22, 1904 – February 18, 1967) was an American theoretical physics, theoretical physicist who served as the director of the Manhattan Project's Los Alamos Laboratory during World ...
and George Volkoff in 1939, using the work of Richard Chace Tolman. Oppenheimer and Volkoff assumed that the
neutron The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
s in a neutron star formed a degenerate cold
Fermi gas A Fermi gas is an idealized model, an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. These statis ...
. They thereby obtained a limiting mass of approximately 0.7 
solar mass The solar mass () is a frequently used unit of mass in astronomy, equal to approximately . It is approximately equal to the mass of the Sun. It is often used to indicate the masses of other stars, as well as stellar clusters, nebulae, galaxie ...
es, which was less than the Chandrasekhar limit for white dwarfs. Oppenheimer and Volkoff's paper notes that "the effect of repulsive forces, i.e., of raising the pressure for a given density above the value given by the Fermi equation of state ... could tend to prevent the collapse." And indeed, the most massive neutron star detected so far, PSR J0952–0607, is estimated to be much heavier than Oppenheimer and Volkoff's TOV limit at ''M''. More realistic models of neutron stars that include
baryon In particle physics, a baryon is a type of composite particle, composite subatomic particle that contains an odd number of valence quarks, conventionally three. proton, Protons and neutron, neutrons are examples of baryons; because baryons are ...
strong force In nuclear physics and particle physics, the strong interaction, also called the strong force or strong nuclear force, is one of the four known fundamental interactions. It confines quarks into protons, neutrons, and other hadron particles, an ...
repulsion predict a neutron star mass limit of 2.2 to 2.9 M. The uncertainty in the value reflects the fact that the
equations of state In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. Most mod ...
for extremely dense matter are not well known.


Applications

In a star less massive than the limit, the gravitational compression is balanced by short-range repulsive neutron–neutron interactions mediated by the strong force and also by the quantum degeneracy pressure of neutrons, preventing collapse. If its mass is above the limit, the star will collapse to some denser form. It could form a
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
, or change composition and be supported in some other way (for example, by quark degeneracy pressure if it becomes a quark star). Because the properties of hypothetical, more exotic forms of degenerate matter are even more poorly known than those of neutron-degenerate matter, most astrophysicists assume, in the absence of evidence to the contrary, that a neutron star above the limit collapses directly into a black hole. A black hole formed by the collapse of an individual star must have mass exceeding the Tolman–Oppenheimer–Volkoff limit. Theory predicts that because of mass loss during
stellar evolution Stellar evolution is the process by which a star changes over the course of time. Depending on the mass of the star, its lifetime can range from a few million years for the most massive to trillions of years for the least massive, which is consi ...
, a black hole formed from an isolated star of solar
metallicity In astronomy, metallicity is the Abundance of the chemical elements, abundance of Chemical element, elements present in an object that are heavier than hydrogen and helium. Most of the normal currently detectable (i.e. non-Dark matter, dark) matt ...
can have a mass of no more than approximately 10
solar mass The solar mass () is a frequently used unit of mass in astronomy, equal to approximately . It is approximately equal to the mass of the Sun. It is often used to indicate the masses of other stars, as well as stellar clusters, nebulae, galaxie ...
es. :Fig. 16 Observationally, because of their large mass, relative faintness, and X-ray spectra, a number of massive objects in X-ray binaries are thought to be stellar black holes. These black hole candidates are estimated to have masses between 3 and 20
solar mass The solar mass () is a frequently used unit of mass in astronomy, equal to approximately . It is approximately equal to the mass of the Sun. It is often used to indicate the masses of other stars, as well as stellar clusters, nebulae, galaxie ...
es.
LIGO The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a large-scale physics experiment and observatory designed to detect cosmic gravitational waves and to develop gravitational-wave observations as an astronomical tool. Prior to LIG ...
has detected black hole mergers involving black holes in the 7.5–50 solar mass range; it is possible – although unlikely – that these black holes were themselves the result of previous mergers. Oppenheimer and Volkoff discounted the influence of heat, stating in reference to work by Landau (1932), 'even t107 degrees... the pressure is determined essentially by the density only and not by the temperature' – yet it has been estimated that temperatures can reach up to approximately >109 K during formation of a neutron star, mergers and binary accretion. Another source of heat and therefore collapse-resisting pressure in neutron stars is 'viscous friction in the presence of differential rotation.' Oppenheimer and Volkoff's calculation of the mass limit of neutron stars also neglected to consider the rotation of neutron stars, however we now know that neutron stars are capable of spinning at much faster rates than were known in Oppenheimer and Volkoff's time. The fastest-spinning neutron star known is PSR J1748-2446ad, rotating at a rate of 716 times per second or 43,000 revolutions per minute, giving a linear (tangential) speed at the surface on the order of 0.24c (i.e., nearly a quarter the speed of light). Star rotation interferes with convective heat loss during supernova collapse, so rotating stars are more likely to collapse directly to form a black hole


List of the least massive black holes


List of objects in mass gap

This list contains objects that may be neutron stars, black holes, quark stars, or other exotic objects. This list is distinct from the list of least massive black holes due to the undetermined nature of these objects, largely because of indeterminate mass, or other poor observation data.


See also

* Bekenstein bound * Oppenheimer–Snyder model * Quark star * Tolman–Oppenheimer–Volkoff equation


References

{{DEFAULTSORT:Tolman-Oppenheimer-Volkoff limit Astrophysics Black holes J. Robert Oppenheimer Neutron stars